Improved Semidefinite Programming Bound on Sizes of Codes

TL;DR

This paper enhances semidefinite programming bounds for binary codes by introducing new linear constraints, resulting in improved upper bounds on code sizes for specific parameters and many new bounds for constant-weight codes.

Contribution

The paper develops new linear constraints added to Schrijver's SDP bound, leading to tighter upper bounds on code sizes and numerous new bounds for constant-weight codes.

Findings

New upper bounds: A(18,8) ≤ 71 and A(19,8) ≤ 131.

23 new bounds for A(n,d,w) with n ≤ 28.

Improved semidefinite programming bounds for binary codes.

Abstract

Let $A(n,d)$ (respectively $A(n,d,w)$) be the maximum possible number of codewords in a binary code (respectively binary constant-weight $w$ code) of length $n$ and minimum Hamming distance at least $d$. By adding new linear constraints to Schrijver's semidefinite programming bound, which is obtained from block-diagonalising the Terwilliger algebra of the Hamming cube, we obtain two new upper bounds on $A(n,d)$, namely $A(18,8) \leq 71$ and $A(19,8) \leq 131$. Twenty three new upper bounds on $A(n,d,w)$ for $n \leq 28$ are also obtained by a similar way.